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Number Theory

The entropy of alpha-continued fractions: numerical results

Profile image of Carlo CarminatiCarlo CarminatiProfile image of Giulio TiozzoGiulio Tiozzo

2010, Nonlinearity

https://doi.org/10.1088/0951-7715/23/10/005
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33 pages

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Abstract

We consider the one-parameter family of interval maps arising from generalized continued fraction expansions known as alpha-continued fractions. For such maps, we perform a numerical study of the behaviour of metric entropy as a function of the parameter. The behaviour of entropy is known to be quite regular for parameters for which a matching condition on the orbits of the endpoints holds. We give a detailed description of the set M where this condition is met: it consists of a countable union of open intervals, corresponding to different combinatorial data, which appear to be arranged in a hierarchical structure. Our experimental data suggest that the complement of M is a proper subset of the set of bounded-type numbers, hence it has measure zero. Furthermore, we give evidence that the entropy on matching intervals is smooth; on the other hand, we can construct points outside of M on which it is not even locally monotone.

Figures (18)
In this paper we will focus on the metric entropy of the T,’s, which by Rohlin’s formula ({I3]) is given by
In this paper we will focus on the metric entropy of the T,’s, which by Rohlin’s formula ({I3]) is given by
Figure 2: Numerical data vs. matching intervals
Figure 2: Numerical data vs. matching intervals
Figure 3: Variance on the interval J = [0.295, 0.304299].
Figure 3: Variance on the interval J = [0.295, 0.304299].
Figure 4: Plot of the standard deviation of the different runs on the Gauss map
Figure 4: Plot of the standard deviation of the different runs on the Gauss map
By diagonalizing the matrices and computing the powers one can compute these value explicitly. In particular,
By diagonalizing the matrices and computing the powers one can compute these value explicitly. In particular,
Figure 5: Size of matchings
Figure 5: Size of matchings
Figure 6: Same picture, in log-log scale.
Figure 6: Same picture, in log-log scale.
Figure 9: Percentage of covering by matching intervals
Figure 9: Percentage of covering by matching intervals
Figure 8: Dependence of the order k = ki + kg of a matching interval on the left endpoint
Figure 8: Dependence of the order k = ki + kg of a matching interval on the left endpoint
Figure 10: The order k; + kz versus the logarithm of the size of the first 10’ matching intervals found.
Figure 10: The order k; + kz versus the logarithm of the size of the first 10’ matching intervals found.
Figure 11: Recursive construction of the matching set
Figure 11: Recursive construction of the matching set
Figure 12: Invariant density for a = 0.338
Figure 12: Invariant density for a = 0.338
A sample of matching intervals found as in section [4
A sample of matching intervals found as in section [4
A chain of adjacent matching intervals (see section |4.
A chain of adjacent matching intervals (see section |4.

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  • Applied Mathematics
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  • Continued Fractions
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